Semi-exactness of the equivariant Kasparov bifunctor
type: Item
instance of: scholarly article
title: Semi-exactness of the equivariant Kasparov bifunctor
zbMATH Open document ID: 0919.19005
DOI: 10.1023/A:1007712719650
author: Mohamed Maghfoul
published in: \(K\)-Theory
publication date: 1999-08-31T00:00:00Z
review text: Let \(G\) be a locally compact group and \(A,B\) be \(G\)-algebras, i.e., \(C^*\)-algebras such that \(G\) acts continuously (in norm) by automorphisms. In the paper ``Equivariant \(KK\)-theory and the Novikov conjecture'' [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)], \textit{G. G. Kasparov} associates to a pair \((A,B)\) of \(\mathbb{Z}/2\mathbb{Z}\)-graduated \(G\)-algebras the abelian group \(KK_G(A,B)\). The aim of this paper is the study of the behaviour of the bifunctor \(KK_G(\cdot,\cdot)\) with respect to exact sequences. Namely, a notion of strong \(K\)-theoretic amenability for \(G\) is defined and all results obtained concerning the behaviour \(KK_G(\cdot,\cdot)\) with respect to exact sequences are generalized to the case \(KK_G(\cdot,\cdot)\) for \(G\) strongly \(K\)-amenable. Moreover, the nuclearity in Skandalis' sense [see \textit{G. Skandalis}, ``Une notion de nucléarité en \(K\)-théorie (d'après J. Cuntz)'', \(K\)-Theory 1, No. 6, 549-573 (1988; Zbl 0653.46065)] is studied in the equivariant context and also the behaviour of the bifunctor \(KK_G\) with respect to exact sequences is given.
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1275466
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: equivariant \(KK\)-theory
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: amenability
zbMATH Keywords: nuclearity
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
full work available at URL: a:1007712719650
OpenAlex ID: W2065920664
description: scientific article; zbMATH DE number 1275466